Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP

Abstract

In this paper we discuss the existence of a normally hyperbolic invariant lamination (NHIL) at the Kirkwood gap $3:1$ for the Restricted Planar Elliptic 3 Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. We also show that the induced dynamics on the NHIL is a partially hyperbolic skew-shift which is of the form \[ f:(ω,I,θ)\to (σω, I+e_0 A_ω(I)\cos(θ+ψ_ω)+\mathcal{O}(e^2_0), θ+Ω_ω(I)+\mathcal{O}(e_0)),\] where $I\in [a,b], θ\in \mathbb T, ω\inΣ=\{0,1\}^\mathbb Z$, the space of sequences of $0,1$'s, $σ:Σ\to Σ$ is the shift in this space, $Ω_ω$ is the shear, $A_ω$ is an amplitude, and $e_0$ is the eccentricity of Jupiter, which is taken as a small parameter. In a companion paper, relying on these skew-shift, we show the existence of stochastic diffusing behavior for Asteroids belonging to the Kirkwood gap provided the eccentricity of Jupiter is $e_0$ small enough. Key ingredients to construct the NHIL are the separatrix map associated to homoclinic channels to a normally hyperbolic invariant cylinder and an isolating block construction. Some of the necessary non-degeneracy conditions are verified numerically.

Figures (18)

• Figure 1: Distribution of asteroids in the Asteroid belt
• Figure 3: The periodic orbits provided by Ansatz \ref{['ans:NHIMCircular:bis']} give rise to a normally hyperbolic invariant cylinder, as stated in Corollary \ref{['coro:NHIMCircular']}. It posseses two transverse homoclinic channels.
• Figure 4: Some (near-)resonant periodic orbits $\lambda_\mathbf{J}$ for different energy values $\mathbf{J}$, or instantaneous eccentricity values $\mathbf{e}(t)$ ($xy$ projection). The location of the Sun and Jupiter are marked with a yellow and brown dot, respectively.
• Figure 5: Solid line: resonant periodic orbit $\lambda_\mathbf{J}$ with energy $\mathbf{J}=-1.456$. Dashed line: trajectory in its associated unstable invariant manifold. As the unstable trajectory precedes clockwise, the horizontal loop of the trajectory ceases to intersect with the section $\{y=0\}$. The loop at an angle $-2\pi/3$ will eventually start intersecting the section.
• Figure 6: Hyperbolic structure on the section $\{y=0\}$ for two different energy levels. The periodic orbit corresponds to $p_0, p_1, p_2, p_4$, fixed points for $\widetilde{\mathcal{P}}$. The stable manifold is colored in blue, and the unstable in red. Points on the symmetry axis have both $y=0$ and $p_x=0 \implies \dot x=0$, so they correspond to symmetric trajectories for the flow. There are four symmetric homoclinic points, denoted $z_1, z_2, z_3, z_4$.

Theorems & Definitions (57)

• Proposition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Theorem 2.5
• Theorem 3.1